#3822083

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,3)(4,8)(10,13)(11,15)(19,23)(22,24)(26,30)(27,33)(28,35)(39,40);(2,3)(6,7)(11,15)(14,16)(25,26,29,30)(27,31,33,38)(28,32,35,36)(34,39,37,40);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24)(25,26)(27,31)(28,36)(29,30)(32,35)(33,38)(34,40)(37,39);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,19,21,23)(18,22,20,24)(25,27,29,33)(26,31,30,38)(28,37,35,34)(32,40,36,39);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,24,23,22)(25,28)(26,32)(27,34)(29,35)(30,36)(31,39)(33,37)(38,40)

Polycyclic presentation:

2^9|b^2=f,c^2=fghi,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3729279);